A = {{2, -1}, {-1, 1}};

B = {{1, -1}, {-1, 2}};

\(f[x_, y_, A_List, B_List] := 

      	{{x, 1 - x} . A . {y, 1 - y}, {x, 1 - x} . B . {y, 1 - y}}\)

f[x, y, A, B] // Expand

f[x, y, A, B] /. y -> 0 // Expand

p1 = ParametricPlot[f[x, y, A, B] /. y -> 0, {x, 0, 1}]

p2 = ParametricPlot[f[x, y, A, B] /. y -> 1, {x, 0, 1}]

p3 = ParametricPlot[f[x, y, A, B] /. x -> 0, {y, 0, 1}]

p4 = ParametricPlot[f[x, y, A, B] /. x -> 1, {y, 0, 1}]

Show[p1, p2, p3, p4]

p5 = ParametricPlot[f[x, y, A, B] /. y -> 1/2, {x, 0, 1}]

p6 = ParametricPlot[f[x, y, A, B] /. x -> 1/2, {y, 0, 1}]

p7 = ParametricPlot[f[x, y, A, B] /. y -> 1/3, {x, 0, 1}]

p8 = ParametricPlot[f[x, y, A, B] /. x -> 1/3, {y, 0, 1}]

p9 = ParametricPlot[f[x, y, A, B] /. y -> 2/3, {x, 0, 1}]

p10 = ParametricPlot[f[x, y, A, B] /. x -> 2/3, {y, 0, 1}]

Show[p1, p2, p3, p4, p5, p6, p7, p8, p9, p10]

Show[
      Table[ParametricPlot[f[x, y, A, B] /. y -> t, {x, 0, 1}], {t, 0, 1, 
          0.1}]]